Is there a version of Girsanov theorem when the volatility is changing?

For example Girsanov theorem states that Radon Nikodym (RN) derivative for a stochastic equation is used to transform the expectation where the sampling is done in one mesaure to an expectation where sampling is done in another measure.

Let's see an example

$dX_t(w) = f(X_t(w))dt + \sigma(X_t(w))dW_t^P(w)$ in P measure.

In P* measure, drift is $f^{*}(X_t(w))$. We multiply the internals of expectation in P measure with RN derivative to get expectation of X in P* measure

Just one small thing. I think you're to do $E^P[X\frac{dP}{dP^*}]$ to change measure from $P$ to $P^*$. Consider this expression in the form of an integral; $\displaystyle \ \ \int_\Omega X (\frac{dP}{dP^*})dP\frac{dP^*}{dP} = E^P[X\frac{dP}{dP^*}]$.
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JaseNov 12 '12 at 14:48

2 Answers
2

The Girsanov theorem applies to any compatible change of measure, including a volatility change. The version you have written above is a simplified version for drift changes only, but if you look in any good stochastic calculus book, you will see that full version just requires that you be able to compute the cross-variation of the two processes.

Come on @Brian B, I know you probably know the answer by heart (haha). Even if not, please include it in your answer it would help the site with another of your good answers... (for later search engines visibility and for later users browsing the site)
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SRKX♦Nov 13 '12 at 0:01

Can you provide a reference for your comment?
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adamNov 13 '12 at 18:27

I actually just checked Karatzas and Shreve without finding the version/result I am thinking of -- did I imagine it? I may withdraw this answer.
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Brian BNov 14 '12 at 4:15

I also remember something about only requiring that your vol be a bounded process with finite quadratic variation in order for the theorem to be valid at this step: $$ d\tilde W_t = (\frac{\mu-r} \sigma )dt + dW_t $$ such that $$ d\tilde W_t $$ is still a brownian motion
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SpeedBootsNov 14 '12 at 20:44

a probability measure assigns relative likelihood to different
trajectories of the Brownian motion. Variance of the Ito process can be
recovered from the shape of a single trajectory (quadratic variation), so it
does not depend on the relative likelihood of the trajectories, hence, does not
depend on the choice of the probability measure.